Question: Let $k$ be a field and $p,q\in k[x]$ two relatively prime polynomials. Prove $p(x)y-q(x)$ is irreducible in $k[x,y]$.
How does one show this? More generally, how does one show that multivariable polynomials are irreducible? In one variable we have access to tools like Gauss's lemma and Eisenstein's criteria, but I do not know any methods applicable to the multivariable case.